6. +
Approximating Functions
f(x) = sin(x)
f(0) = 1, f’(0) = 1, f’’(0) = 0, f’’’(0) = -1,…
What is f(1)? i.e . What is sin(1)?
7. +
Famous Dead People
James Gregory (1671)
Brook Taylor (1712)
Colin Maclaurin (1698-1746)
Joseph-Louis Lagrange (1736-1813)
Augustin-Louis Cauchy (1789-1857)
8. +
Approximations
Linear Approximation
f (x) = f (a) + f ʹ′(a)(x − a) + R1 (x)(x − a)
R1 (x)(x − a) = f (x) − f (a) − f ʹ′(x − a)
Quadratic Approximation
f ʹ′ (a)
€ f (x) = f (a) + f ʹ′(a)(x − a) + (x − a) 2 + R2 (x)(x − a) 2
2 f ʹ′ (a)
2
R2 (x)(x − a) = f (x) − f (a) − f ʹ′(a)(x − a) − (x − a) 2
2
€
9. +
Taylor’s Theorem
Letk≥1 be an integer and f : R →R be k
times differentiable at a ∈ R .
Then there exists a function R : R →R such that
k
f ʹ′ (a) f k (a)
f (x) = f (a) − f ʹ′(a)(x − a) + (x − a) 2 + ...+ (x − a) k + Rk (x)(x − a) k
2! k!
€
€
Note: Taylor Polynomial of degree k is:
€
f ʹ′ (a) f k (a)
Pk (x) = f (a) − f ʹ′(a)(x − a) + (x − a) 2 + ...+ (x − a) k
2! k!
10. +
Works for Linear
Approximations
f (x) = c 0 + c1 (x)
f (a) = c 0 + c1 (a)
f ʹ′(a) = c1 f (x)(a)fʹ′= c + c (x − a)
f = f (a) = c (a)
+ 0 11
1
€ f (x) = f (a) + c1€ € € a)
(x −
€ f (x) = c 0 + c1 (a) + c1 (x − a) = c 0 + c1 (x)
€
€
€
11. +
Works for Quadratic
Approximations
f (x) = c 0 + c1 (x) + c 2 (x 2 )
f (a) = c 0 + c1 (a) + c 2 (a 2 )
f ʹ′(a) = c1 + 2c 2 (a)
f ʹ′ (a) = 2c 2
€ = c + c (a) + c (a 2 ) + [c + 2c (a)](x − a) + 2c 2 [x − a]2 =
€
f (x) 0 1 2 1 2
€ 2
c 0€ c1(a) + c 2 (a 2 ) + c1(x − a) + 2c 2 (x − a) + c 2 (x) 2 − c 2 (2ax) + c 2 (a) 2 =
+
f (x) = c 0 + c1 (x) + c 2 (x 2 )
€
17. +
Implications
Any smooth functions with all the same derivatives
at a point MUST be the same function!
18. + Proof: If f and g are smooth functions that agree
over some interval, they MUST be the same function
Let f and g be two smooth functions that agree for some open
interval (a,b), but not over all of R
Define h as the difference, f – g, and note that h is smooth, being the
difference of two smooth functions. Also h=0 on (a,b), but h≠0 at
other points in R
Without loss of generality, we will form S, the set of all x>a, such
that f(x)≠0
Note that a is a lower bound for this set, S, and being a subset of R,
S is complete so S has a real greatest lower bound, call it c.
c, being a greatest lower bound of S, is also an element of S, since S
is closed
Now we see that h=0 on (a,c), but h≠0 at c. So, h is discontinuous at
c, but then h cannot be smooth
Thus we have reached a contradiction, and so f and g must agree
everywhere!
19. +
Suppose f(x) can be rewritten as a
power series…
f (x) = c 0 + c1 (x − a) + c 2 (x − a) 2 + ...+ c n (x − a) n
c 0 = f (a)
f ʹ′(x) = c1 + 2c 2 (x − a) + 3c 3 (x f − a) 2 + ...+ nc n (x − a) n −1
k
(a)
ck =
€ k!
c1 = f ʹ′(a)
€ f ʹ′ (x) = 2c 2 + 3∗2c 3 (x − € + 4 ∗ 3c 4 (x − a) 2 + ...+ n ∗(n −1)c n (x − a) n −2
a)
€
f ʹ′ (a)
c2 =
€ 2!
€ f k (a)
ck =
k!
€
20. +
Entirety (Analytic Functions)
A function f(x) is said to be entire if it is equal to its
Taylor Series everywhere
Entire Not Entire
sin(x) log(1+x)
21. +
Proof: sin(x) is entire
Maclaurin Series
sin(0)=1
sin’(0)=0
∞
sin’(0)=-1 (−1) n 2n +1
sin(x) = ∑ x
sin’(0)=0
n =0 (2n +1)!
sin’(0)=1
sin’(0)=0
sin’(0)=-1
… etc. €
22. +
Proof: sin(x) is entire
∞
(−1) n 2n +1
sin(x) = ∑ x
n =0 (2n +1)!
Lagrange formula for the remainder:
Let f : R →R be k+1 times differentiable on
(a,x) and continuous on [a,x]. Then
f k +1 (z) k +1
Rk (x) = (x − a)
(k +1)!
€ for some z in (x,a)
€
23. +
Proof: sin(x) is entire
First, sin(x)
is continuous and infinitely
differentiable over all of R
If we look at the Taylor Polynomial of degree k
f k +1 (z) k +1
Rk (x) = (x − a)
(k +1)!
Note though f k +1 (z) ≤ 1 for all z in R
k +1
(x − a)
Rk (x) ≤
€ (k +1)!
€
€
24. +
Proof: sin(x) is entire
However, as k goes to infinity, we see Rk (x) ≤ 0
Applyingthe Squeeze Theorem to our original
€
equation, we obtain that as k goes to infinity
f (x) = Tk (x)
and thus sin(x) is complete
€
25. +
Maclaurin Series Examples
∞ ∞
xn xn
log(1 − x) = −∑ log(1+ x) = ∑ (−1) n +1
n =1 n! n =1
n!
∞ ∞
1 (−1) n (2n)!
= ∑ xn 1+ x = ∑ xn
1 − x n =0 n =0
(1 − 2n)(n!) 2 (4) n
∞
€ xn€
ex = ∑
n =0 n!
∞
€ ∞
€
(−1) n
(−1) n 2n
sin(x) = ∑ x 2n +1 cos(x) = ∑ x
n =0
(2n +1)! n =0 (2n)!
€ ix
Note: e = cos(x) + isin(x)
€ €
26. +
Applications
Physics
Special Relativity Equation
Fermat’s Principle (Optics)
Resistivity of Wires
Electric Dipoles
Periods of Pendulums
Surveying (Curvature of the Eart)
27. +
Special Relativity
m0
m= KE = mc 2 − m0c 2
1− v2 c2
m0c 2 ⎡⎛ v 2 ⎞ −1/ 2 ⎤
KE = − m0c 2 = m0c 2 ⎢⎜1 − 2 ⎟ −1⎥
2
1−v c 2 € ⎢⎝ c ⎠
⎣ ⎥
⎦
€
If v ≤ 100 m/s
€
Then according to Taylor’s Inequality
1 3m0c 2 100 4 −10
R1 (x) ≤ 2 2 4 < (4.17 × 10 )m0
2 4(1 −100 /c ) c
